About 700 police officers and support staff were involved in the early morning operation at addresses in the UK, Belgium and the Netherlands.
Police say the raids were one of the largest operations against animal rights extremists in the UK.
The 15 men and 17 women arrested are being questioned at several undisclosed police stations.
Eight people were arrested in raids at seven addresses in Hampshire.
There were 21 other addresses raided in Berkshire, Kent, Lancashire, London, Merseyside, Northumbria, Oxford, Surrey, Sussex, Worcestershire, Yorkshire, Glasgow and Aberdare, south Wales from 0530 BST.
Two unnamed locations in the Netherlands and one in Belgium were also raided but no arrests were made.
Although the vast majority of animal rights protesters campaign lawfully, a small minority seeks to force change through criminal action
ACC Adrian Leppard
Police say the "substantial operation" targeted burglary, conspiracy to blackmail, and offences against animal research operations.
The extremists' targets included Huntingdon Life Sciences in Cambridgeshire, officers said.
Assistant Chief Constable Adrian Leppard, of Kent Police, who commanded the operation, said: "In recent years, animal rights extremists have conducted sustained campaigns of harassment and intimidation against the animal research industry, seeking to achieve their objectives by creating a climate of fear.
"Although the vast majority of animal rights protesters campaign lawfully, a small minority seeks to force change through criminal action."
Aisling Burnand, chief executive of the BioIndustry Association (BIA), said: "News of today's operation will act as a great fillip to the medical research community across Europe.
"Recent efforts of the police, government and judiciary have resulted in a significant decline in the incidence of animal rights extremism and are much welcomed by the UK's bioscience community."
Mr Leppard said the operation focused on the south-east of England, because that was where extremist activity was concentrated.
Action was being taken to minimise any disruption caused by the raids, he added.
One of the locations raided was Freshfields Animal Rescue Centre in Ince Blundell, Merseyside.
Centre manager Dave Calendar, who says they have "nothing to hide", accused the police of "heavy-handed tactics".
Merseyside Police rejected claims that any injuries had been caused during the raid.
The operation is ongoing and no one has yet been charged.